Abeysekera was born on 3 June 1930 in Ratmale near Matara in Southern Sri Lanka. He was educated at Nalanda College Colombo. Some of his notable classmates at Nalanda College were Dr Gunadasa Amarasekara, Stanley Jayasinghe, Dr Harischandra Wijayatunga, Dr Hudson Silva, Hon. Rupa Karunathilake, Hon. Dr Dharmasena Attygalle, Dr Henry Jayasena, Bernie Wijesekera. Siri Perera was Abeysekera's Sinhala language teacher at Nalanda.
He was discovered by the children's radio programme Ḷamā Piṭiya ("Children's Field") hosted by Siri Aiya (also known as U.A.S. Perera - Siri Perera QC) and broadcast over Radio Ceylon, the oldest radio station in South Asia. Ḷamā Piṭiya was a showcase for young talents, and Abeysekera performed with his poems on the Radio Ceylon programme in the 1940s. He was an instant hit with the general public and remained a pop icon until his death.
His younger brother Daya Abeysekara was a renowned journalist and an actor. He died on 29 March 2019 at the age of 80. Funeral will be held on 31 March 2019 at Kanatte Cemetery.